Good afternoon, Math Kangaroo students. Welcome to Webinar 7. This is the Level 5-6 series. I'm Sarah Segee. Most of you have been here, so you're familiar with me in the format. I have my chat open, so if you have questions as you're solving problems, you can try to send them to me in the chat. I unfortunately can't answer every single inquiry, but I do do my best. I also have polls for many of today's questions. The topic today is logical thinking problems. So we have several different types of logic problems that we're going to solve today. There's usually at least one or two logic problems on every Math Kangaroo contest, and logic problems serve us well not only in math, but in other areas of our academic striving as well. Like, if you were trying to write an argumentative essay and your logic was not following a good order, it would be very confusing for people to try to read and agree or disagree with your points. So logic is something we're going to use very frequently. Logic problems that we'll see today, they might assume that you know a few things about mathematics, such as evens and odds, or different types of sets of numbers, integers, or things like that. They might require you to know days of the week or things like that. So we do assume some other background knowledge in our logic problems. So here's our first warm-up problem. Our teaching assistant is not going to be able to make it today. He was participating in a competition, so he's on an airplane. So it will be just me. Hopefully I'll be able to get to everyone's questions today. You're at a fork in the road in which one direction leads to the city of lies, where everyone always lies, and the other to the city of truth, where everyone always tells the truth. There's a person at the fork who lives in one of the cities, but you're not sure which one. What question could you ask the person to find out which road leads to the city of truth? There's no poll for this one, but go ahead and see if you can post some questions into the chat, and we'll see if you get to a question that will give us an answer. Thank you, I'm getting some responses in the chat. I'm going to read some of them aloud. Does that way lead to the City of Lies? Which road is not the City of Lies? Can ask the person if they are lying. Ah, somebody has it. Where do you live? So yeah, I like the question, where do you live? Let's see why. So where do you live? If I ask a liar, where do you live? The liar will point to the City of Truth, correct? If I ask a truth teller, where do you live? If I ask a truth teller, where do you live? They're going to point to the City of Truth. So no matter who I ask, I'm going to get somebody. If I ask, where do you live? They'll both point to the City of Truth. And that way, I will know in which way is the direction of the City of Truth and in which way is the direction to the City of Lies. Yeah, so some tricky reasoning there, but that is excellent. So in all of the problems today, you'll notice that reading and understanding the problem is going to be of critical importance. So be very, very careful with your reading. Sometimes you'll want to read one sentence at a time and take notes. What does that sentence say? What information did I get just from that part? And sometimes after you have all of that information, you're going to be able to come up with your plan on how to solve the problem. Well, also, if there's any optional, like a second way to solve it, that could be a good way to check and reflect and see if your answer makes sense. Sometimes a second solution by another method with the same solution, that helps a lot. OK, so let's do our best today with these really fun logical reasoning problems. Oh, my goodness. So in math, logic refers to the methods of reasoning to determine whether an argument is true. For example, the process of elimination. Does B meet a given condition? Like if I put an equal sign in the equation, is it really equal? Is 5 plus 3, 8? That can be a true or false, right? 5 plus 3 is 8. It is true. But I could also write an incorrect equation, and it could be false. And one place that you will often see these kinds of things is if I do like greater than and less than, right? If I say A is greater than B, and then I'm going to do, and then B is greater than C, then I'll know that A is greater than C. That's reasoning. And you may have seen that many times in your math problems, things like that. OK, so if at any point something is false, you may have to back up. If something is false, you may have to back up and change your assumptions. That's very common in logical reasoning problems. So why do we do logical reasoning problems? They help us make good decisions and avoid common logical mistakes. So here's a logical mistake. This is a little kitten, and it says, all dogs have four legs and fur. I have four legs and fur. I must be a dog. See the logical issue there? Just because dogs have four legs and fur does not mean everything with four legs and fur is a dog. Some of you may know I'm a biologist. So one of the things that a Nobel Prize winning biologist said to me when I was in a class is, just because gum sticks to your shoe does not mean your shoe is designed to pick up gum. Your shoe is designed to protect your foot, but not to pick up gum. That's just a side effect. So you can think about these different kinds of reasoning problems. A lot of times people say, oh, well, gum always sticks to my shoe. That must be what shoes are for. No, we don't want gum stuck to our shoes. Some examples that we'll do today that you might find on Math Kangaroo or other math contests are cryptorhythms, where we're substituting letters or symbols for numbers, certainty problems. These can also be presented as, it's called the pigeonhole principle. Like if you have so many socks in a drawer and you're picking them out in the dark. Then diagram problems, where you have things that could be overlapping circles. Whodunnits, these are also more technically called like one-to-one correspondence problems. Those will be like, you have four students doing four different sports, wearing four different jerseys. And then the truth and lie, like the example problem that we had for our warmup. So hopefully you'll have a lot of fun today. Do something a little bit different than we've done in the other lessons. In this sum, the same letters represent the same digits and different letters represent different digits. What digit does the letter X represent? Remember I said, you'll need to know some math, right? You can't fill this in unless you know about what Z could be. I'll give a minute or two, and then I do have polls for a lot of questions today. So you'll be able to answer in the poll. Anybody else want to put an answer into the poll? Remember, we're trying to find what X represents. Okay, we'll end it here. We've had most of you participate in the poll, but not all. You can see that the most common answer, 60 percent of you say that X is a six. That is correct. Let's take a look. I'm going to ask that students do not annotate on the screen. Okay, so when you're adding a two-digit number, the largest two-digit number you could possibly add would be 99. So the three-digit number that we would have to get would start with a one. It would be in the 100s. So this has to be 1, 1, 1. If that's 1, 1, 1, this has to be either 88 or 99. Let's try making it 99. So if we take 111 minus 99, we are going to get that the difference is 12. So that means that X plus X has to equal 12. So of course, X equals six. Now, that's a great way to solve this problem. In a bag, there are three green apples, five yellow apples, seven green pears, and two yellow pears. Simon randomly takes fruit out of the bag one by one. How many pieces of fruit must he take out in order to be sure that he has at least one apple and one pear of the same color? Now, read carefully. What does it mean, one apple and one pear of the same color? So they would have to both be green or both be yellow. There's also an interesting thing. He's taking them out without looking, and we have to be sure. So this is the certainty problem. Yeah, there is a pole. I was giving students just a minute to take a look at the problem nice and big, because sometimes those poles are small. Here it is. OK, anybody else want to put their answer in the poll? Somebody says the poll was not working. It worked for me. It's worked for 22 people, so I think it's working. OK. So here we go. Not everyone answered the poll, but we have a little bit of a lead for 13 items that you need to pick, but also 10 and 11 have been some pretty popular choices. So let's take a look. The big consideration is that we need, uh-oh, sorry about that. That's my mistake. Sometimes I click on the wrong little bit on my screen. OK. I apologize. So when we're dealing with a certainty problem, in order to be certain that you are going to get what you want, you have to first assume that you are very unlucky and that you never get what you want, right? So if you assume you're unlucky and everything goes poorly, then you'll be assured that you get it. Hopefully that makes a little bit of sense. All right. So if I'm really unlucky, remember I want the same color apple and same color pear. So if I'm really unlucky, I'm going to draw out the five yellow apples, and then I'm going to draw out the seven green pears. All right, so none of those match in color. So that's the most unlucky thing I can do, is draw out five yellow apples and seven green pears. Now the next piece of fruit I take will be either a yellow pear or it will be a green apple. A green apple. So if I draw out either a yellow pear, it will match here. If I draw out the green apple, it will match there. So the 12 items were drawn out in an unlucky fashion, but the 13th item has to be a color match to one of the others that I've drawn out. So 13 is the correct answer to make sure I am certain to have a matching color in a different fruit. Kind of fun problems, right? 66 cats signed up for the contest Miss Cat 2013. After the first round, 21 were eliminated because they failed to catch mice. 27 cats out of those that remained in the contest had stripes, and 32 of them had one black ear. All the striped cats with one black ear got to the final. What is the minimum number of finalists? Again, there is a poll, but I'll give you a few seconds to read the problem a couple of times before I launch it. Alright, so most of you have now answered the poll, thank you for that. And we have pretty much, kind of a, pretty close between 14 and 5. So let's take a look at the way Math Kangaroo has solved this problem. I did have a student asking, why is it asking for the minimum number of finalists? It says all the striped cats with one black ear got to the final, but it doesn't say perhaps some other cats did as well. So that's why this is the minimum number of finalists. For all we know, maybe there's some white cats in the finals, we don't know. Okay? It's a 66 cats signed up for the contest, and after the first round, 21 were eliminated because they failed to catch mice. So we had 66, and then 21 of those were knocked out. So that is 45. And it says of those 45, 27 had stripes, and 32 of them had one black ear. So if we count 27 with the stripes, and we count 32 with a black ear, we'll call it a blear for black ear, that is 59, and 59 is greater than 45. So some of them have both a stripe and black ear. If we take the 59 minus 45, that is going to tell us how many of them overlap, which is 14. Some of you might try to solve this using a Venn diagram. So you would have a Venn diagram system where you know that the total is 45. You know that in the stripe category, you have 27, and you know that in the black ear category, you have 52, but, ah, sorry, you have 32. You ever have one of those days? So then you have to figure out what is this overlap so that your total is 45, and you would do the exact same thing. You would add them up and subtract. Three friends, Smith, Roberts, and Farrell, each have one of these professions, doctor, engineer, and musician. Each one has a different profession. The doctor does not have a sister nor a brother. He is the youngest of the three friends. By he, they mean the doctor because that's what the sentence before states was the subject. Farrell is older than the engineer, and is married to Smith's sister. What are the names of the doctor, the engineer, and the musician in that order? So a couple of things. Here the subject is the doctor, and when it says he, that's who it must be referring to, right? There's no other logical choice for that. It says Farrell is older than the engineer, and is married. So Farrell is older, and Farrell is married to Smith's sister. I'll give you a few moments to figure this out. I do have a poll. Does anybody else want to reply in the poll? Still have a few of you who have not. Remember, math kangaroo contests do not subtract for wrong answers. So if you run out of time and you haven't answered every problem, it's worth it to fill in the bubbles, because you might guess correctly and get some extra points. All right. I'll give you 20 seconds before I close the poll. All right, well, of those who answered it, we do have the majority of you getting them in the correct order. So that's really, really good. Like I said, not everyone answered. And there is a pretty good proportion of you who either didn't answer or got it incorrect. So let's take a look at it very carefully. This is the type of problem that's called one-to-one correspondence or whodunnits. Okay, and for me, I love to make tables for my one-to-one correspondence problems because I know it's gonna be one person is each of these professions. So I'll be able to put check marks in the boxes that are in the column and the row for that combination. And then I can put Xs where it doesn't work. And by process of yes, affirmative, or process of elimination where, oh, that one can't be, I think I'll get there. All right, so we know it's one profession per person, and I set up my little table. The doctor does not have a sister nor a brother. Well, that doesn't help us with this, but we know that the doctor doesn't have a sister or brother. The doctor is the youngest of the three friends. So doctor, I'm gonna write here, doctor has no siblings and is the youngest. This may not seem to get us an answer yet, but it'll help, I promise. Feral is older than the engineer. Okay, so now I know feral is not the engineer. I can cross that off. If it's gonna put them in order of age, I would have feral, the engineer. I'm having a little trouble with my, my coordination with writing, and then the doctor, right? So by age, feral is the older than the engineer, and we know that the doctor is the youngest. So feral is older than the engineer, and he's married to Smith. Smith has a sister. So feral is not the engineer. He's married to Smith's sister. Smith has a sister. So Smith is not the doctor because the doctor has no siblings. We also know, so Smith has a sister, and we know that feral is older than the engineer. Feral is not an engineer. We know that feral is not, we know that the, sorry. We know that Smith is not the engineer as well because we wouldn't say older than the engineer and is married to Smith's sister. Hang on. So Farrell, sorry about that, I just had to double check my notes. Farrell is older than the engineer, so Farrell is not the engineer. And Farrell is married to Smith's sister. Farrell would not be married to his own sister, right? So we know, therefore, that Farrell is not the doctor, and then we know that Farrell is the musician. That leaves us with Roberts is the doctor, and the only other profession is the engineer. Let's see if this works. The doctor does not have a sister or a brother, but we know that Smith does. We know the doctor is not Smith, so not Smith. Because Smith does have a sister and the doctor does not, so Smith is not the doctor. Farrell is older, so Farrell cannot be the doctor because the doctor is the youngest, so Roberts must be the doctor. Okay, so now we know that the order that they would go in, doctor, engineer, musician, is this order. So it's Roberts, Smith, Farrell, F. Sorry about my hesitation there. Sometimes even I have to double check my notes. So it's really good to do these problems over and over again to really get into good, logical, clear thinking patterns. Oh, I had a table there. Could have used that one. Okay. When it is raining, the cat stays in the room or in the basement. When the cat stays in the room, the mouse is in the foyer and the cheese is in the refrigerator. When the cheese is on the table and the cat stays in the basement, the mouse is in the room. Right now, it is raining and the cheese is on the table. So for sure, A, the cat is in the room, or B, the cat is in the room and the mouse is in the foyer, or C, the mouse is in the foyer, or D, the cat is in the basement and the mouse is in the room, or E, the situation is impossible. If you don't know where to start on this problem, I would suggest read the answer choice and test it against the four conditions above. See if it could be true. If not, you can cross it out as being false. And do that for all of them. Okay, this one was actually pretty good responsiveness, most of you answered. And most of you have the same answer, D, the cat is in the basement and the mouse is in the room. That is correct. Let's take a look at it together. We know from four that right now it is raining and the cheese is on the table. So if we look at the first one, when it is raining, the cat stays in the room or the basement. So we can have the cat in the room or the basement. When the cat stays in the room, we know that the mouse will be in the foyer and the cheese is in the fridge. Okay, from number three, when the cheese is on the table, the cat stays in the basement. This would be cheese on the table. All right, so right now it is raining. So the cat has to be in the room or the basement and the cheese is on the table. If the cheese is on the table, the cat must be in the basement. Right? Let's see what it says. When the cat stays in the basement, the mouse is in the room. So we know mouse is in the room. So now we know the cat is in the basement, the mouse is in the room, the cheese is on the table. So the cat is in the room, that's false, right? The cat is in the basement. Cat is not in the room. Is the mouse in the foyer? No, the mouse is in the room. The cat is in the basement and the mouse is in the room, that's true. The situation is possible because we figured out that the cat will be in the basement and the mouse will be in the room. So we use our different clues and we can figure it out. Fourteen people are seated at a round table. Each person is either a liar or tells the truth. Everybody says, both my neighbors are liars. So they mean the neighbor on the right and the neighbor on the left. What's the maximum number of liars at the table? So something that they assumed you knew is what does it mean by neighbors? If you're sitting at a table, your neighbors are on either side of you. We do have a poll. Bye. All right, most of you have answered. So I will share the results of the poll. And we have pretty much a tie for seven and nine. So let's see which way it really goes. It says that we have 14 people seated at a round table. Okay, so let's make a round table. Why not, right? Here's a round table. Each person, and we have 14 people here. So I'm gonna say one, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13. I wanna move this over a little bit to make room for 14 and one. Okay, each person is either a liar or tells the truth. And everybody says, both my neighbors are liar. And I want to maximize the number of liars. So let's put in a liar. If that person says, both my neighbors are liars, we know that person is lying. So both the neighbors cannot be liars, but one could be a truth teller and one could be a liar. When the truth teller in position number two says, both my neighbors are liars, that is the truth. When the liar in number three says it, we know that it's not the truth, but we want to maximize the number of liars around the circle. So we'll make one neighbor a liar and one neighbor a truth teller. Then when number four says, both my neighbors are liars, that's a lie, he is sitting next to one liar. So we better put a truth teller next to him. The truth teller will correctly say both are liars. And you can notice that we have this pattern going around. So the liar says both, but we're gonna just make one of them a liar, trying to make as many liars as possible. Now we have a truth teller. We'll do the same liar, liar, a truth teller. The truth teller will say there are two liars. Now, if I'm spot number 13, I put a liar and that liar says both my neighbors are liars, that would be a truth. So that would break the rules. So I have to put a truth teller here because both neighbors are liars. So now if we count up the number of liars, we have one, I'll do another color. We have one, two, three, four, five, six, seven, eight, nine liars is the maximum number of liars. If you have a little extra time, you might try this problem thinking what is the minimum number of liars at the table? It could be another way to look at this problem and to have fun solving it. All right, we have time for some bonus problems. A magical kingdom is inhabited by dragons with six, seven, and eight heads. Those with seven heads always lie and those with six or eight heads always tell the truth. One day, four dragons met. The blue dragon said, together we have 28 heads. The green dragon said, together we have 27 heads. The yellow dragon said, together we have 26 heads and the red dragons said, little typo there, the red dragon said, together we have 25 heads. What color was the dragon that did not lie? This is a tricky five point question and number 30 at the very end of a contest. Okay, I will end the poll, most of you have answered. And I'll share the results. There is one answer that is more common than the others, but it's not 50%. So 40% of you think that it's green, but if we add up the yellow and impossible to determine, that's over 40%. So maybe we're a little bit unsure of this question. So let's take a look, there are a few things, a few very important clues in this question. So the first is we have four dragons. So there are four dragons that we're talking about, right? If there was more than one truth-telling dragon, then we would get two of the same answers, right? But we have four different answers. It also says, what was the color of the dragon that did not lie? So if one dragon does not lie, then three dragons lied. So by two different scenarios, two different reasoning processes, the fact that we don't have two of the same answer with two truth-tellers, we get two of the same answer. And the fact that it says one dragon did not lie, we know that three are liars. Well, it says that those with seven heads always lie. So we have seven times three, that's 21 heads on the three lying dragons. Dragons. So there's one truthful dragon, truthful dragon. And how many heads do truthful dragons have? Truthful dragons have six or eight heads. So we'd need to take 21 plus six equals 27 or 21 plus eight equals 29. Right? Well, is one of these answers 27? One of the things that has been said by the truth-teller dragon? The truth-teller dragon is this green dragon who has said together we have 27 heads. That does match. So the green dragon was the truth-teller who said together we have 27 heads. Hopefully that makes sense now that you have seen how I've solved it. So let's have just a little wrap up because I know that this is a fun lesson. Okay, so some different ways that you can solve these logic problems is you can draw diagrams. With the people around the table, I literally drew the circle with 14 places in order to figure it out. That was the fastest way for me to figure it out. You can make a table. With the doctors and the engineers, we made a table. We can do some guessing and checking. That also works. You might need a diagram like the Venn diagram. So working with these logic problems helps enhance our reasoning ability. And again, that's important for the math problems and also for everyday life. A lot of times when you're a scientist like I am, you'll have to determine is something just a coincidence or does it cause it? There's a difference logically thinking between a causal relationship and a coincidence. There's a difference between seeing a whole bunch of clues that agree with your hypothesis or actually having experiments that prove your hypothesis. There's kind of a difference, right? If I just observe everything, I could observe a rainbow in the sky and come up with some explanations for it. But in order to prove how the rainbow gets into sky, I actually have to be able to come up with a situation where I can create my own rainbow and say, oh, this creates a rainbow. Do I have these conditions available in the rain when the sun is shining through the raindrops? So there's a difference between things that go together and things that prove each other. And remember, put all of the pieces together in your logical thinking problems. One of the best strategies will be to read sentence by sentence. We had one in a different level that I was teaching today that said there are certain animals in the park. There are 15 animals in the park and 10 are not cows. It said precisely 10 are not cows. Well, if there are 15 animals and 10 are not cows, then the other five are in the group that are cows. So sometimes you'll be able to use the opposite, right? If my light is not off, then my light is on because there's only two states for the light, on or off. Plus it has a dimmer switch. So I hope everyone's enjoyed the logic problems. I think they're challenging and a lot of fun and they really stretch our brains in a different way. I hope you enjoyed President's Day. I know most of you have the day off tomorrow. I will see everybody back here again next week. And remember, you can always watch the videos if you wanna see these problems again, try them out again. Or if you missed any of the webinars, go back to the videos to watch them. Use past contests to practice timing yourself and using all the different types of problems in a row and figuring out what you need to use. I have a student who asked to see the warmup problem one more time because they arrived late. I'll do that right now. Otherwise, we'll see everybody later. There you go. This is the warmup problem. You're at a fork in the road in which one direction leads to the cities of lies where everyone always lies and the other to the city of truth where everyone always tells the truth. There's a person at the fork who lives in one of the cities but you're not sure which one. What question could you ask the person to find out which road leads to the city of truth? You want to send a question that you think you could ask, you can put that into the chat. So that's pretty close. I have one student say, in which direction do you come from? Meaning, in which direction do you live? And that would be a great question to ask. So if we ask, in which direction do you live? Let's see if I can get this working. The liar will point to the truth, because the liar is going to not tell you where they really come from. And the truth teller will also point to the city of truth, because the truth pointer is going to tell you the correct place. So yes, if you ask in which direction do you live, then you'll be able to figure out which road leads to the city of truth. OK? So if you were late, now you have the warm-up problem. Have a wonderful week, everybody. Bye-bye.

logical thinking

problem-solving

math competitions

reasoning abilities

cryptarithms

pigeonhole principle

Venn diagrams

dragons truth

city of truth

Sarah Segee